Integrand size = 16, antiderivative size = 155 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {10 a^2 \left (a b+\left (b^2-2 a c\right ) x\right )}{c \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 a^3 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:
1/3*x^5*(b*x+2*a)/(-4*a*c+b^2)/(c*x^2+b*x+a)^3-5/3*a*x^3*(b*x+2*a)/(-4*a*c +b^2)^2/(c*x^2+b*x+a)^2-10*a^2*(a*b+(-2*a*c+b^2)*x)/c/(-4*a*c+b^2)^3/(c*x^ 2+b*x+a)+40*a^3*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(155)=310\).
Time = 0.16 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.03 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\frac {b^7-12 a b^5 c+48 a^2 b^3 c^2-59 a^3 b c^3-4 b^6 c x+33 a b^4 c^2 x-72 a^2 b^2 c^3 x+26 a^3 c^4 x}{3 c^5 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {-b^7+12 a b^5 c-48 a^2 b^3 c^2+74 a^3 b c^3+b^6 c x-12 a b^4 c^2 x+48 a^2 b^2 c^3 x-44 a^3 c^4 x}{c^4 \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {b^6 x+a b^4 (b-6 c x)+a^3 c^2 (5 b-2 c x)+a^2 b^2 c (-5 b+9 c x)}{3 c^5 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {40 a^3 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}} \] Input:
Integrate[x^6/(a + b*x + c*x^2)^4,x]
Output:
(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 59*a^3*b*c^3 - 4*b^6*c*x + 33*a*b^4*c ^2*x - 72*a^2*b^2*c^3*x + 26*a^3*c^4*x)/(3*c^5*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (-b^7 + 12*a*b^5*c - 48*a^2*b^3*c^2 + 74*a^3*b*c^3 + b^6*c*x - 12*a*b^4*c^2*x + 48*a^2*b^2*c^3*x - 44*a^3*c^4*x)/(c^4*(-b^2 + 4*a*c)^3*( a + x*(b + c*x))) + (b^6*x + a*b^4*(b - 6*c*x) + a^3*c^2*(5*b - 2*c*x) + a ^2*b^2*c*(-5*b + 9*c*x))/(3*c^5*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (40* a^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2)
Time = 0.53 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1153, 1153, 1153, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {10 a \int \frac {x^4}{\left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {10 a \left (\frac {x^3 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a \int \frac {x^2}{\left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {10 a \left (\frac {x^3 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a \left (\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 a \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {10 a \left (\frac {x^3 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a \left (\frac {4 a \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}+\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {10 a \left (\frac {x^3 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a \left (\frac {4 a \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}\) |
Input:
Int[x^6/(a + b*x + c*x^2)^4,x]
Output:
(x^5*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (10*a*((x^3*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*a*((x*(2*a + b*x))/((b ^2 - 4*a*c)*(a + b*x + c*x^2)) + (4*a*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c ]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)))/(3*(b^2 - 4*a*c))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(530\) vs. \(2(147)=294\).
Time = 0.84 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.43
method | result | size |
default | \(\frac {-\frac {\left (44 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{5}}{c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {b \left (14 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{4}}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (160 a^{4} c^{4}-286 a^{3} b^{2} c^{3}+12 a^{2} b^{4} c^{2}+7 a \,b^{6} c -b^{8}\right ) x^{3}}{3 c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b a \left (16 a^{3} c^{3}+53 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) x^{2}}{c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} \left (20 a^{3} c^{3}-66 a^{2} b^{2} c^{2}+13 a \,b^{4} c -b^{6}\right ) x}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {\left (66 a^{2} c^{2}-13 c a \,b^{2}+b^{4}\right ) a^{3} b}{3 c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {40 a^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\) | \(531\) |
risch | \(\frac {-\frac {\left (44 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{5}}{c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {b \left (14 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{4}}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (160 a^{4} c^{4}-286 a^{3} b^{2} c^{3}+12 a^{2} b^{4} c^{2}+7 a \,b^{6} c -b^{8}\right ) x^{3}}{3 c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b a \left (16 a^{3} c^{3}+53 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) x^{2}}{c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} \left (20 a^{3} c^{3}-66 a^{2} b^{2} c^{2}+13 a \,b^{4} c -b^{6}\right ) x}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {\left (66 a^{2} c^{2}-13 c a \,b^{2}+b^{4}\right ) a^{3} b}{3 c^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {20 a^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 a^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\) | \(650\) |
Input:
int(x^6/(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
(-(44*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c/(64*a^3*c^3-48*a^2*b^2*c^2+ 12*a*b^4*c-b^6)*x^5-b*(14*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(64*a^3*c ^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2*x^4-1/3/c^3*(160*a^4*c^4-286*a^3*b^2 *c^3+12*a^2*b^4*c^2+7*a*b^6*c-b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b ^6)*x^3+b/c^3*a*(16*a^3*c^3+53*a^2*b^2*c^2-12*a*b^4*c+b^6)/(64*a^3*c^3-48* a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-a^2*(20*a^3*c^3-66*a^2*b^2*c^2+13*a*b^4*c- b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3*x+1/3*(66*a^2*c^2-13*a *b^2*c+b^4)*a^3*b/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b *x+a)^3+40*a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2 )*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (147) = 294\).
Time = 0.12 (sec) , antiderivative size = 1675, normalized size of antiderivative = 10.81 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(c*x^2+b*x+a)^4,x, algorithm="fricas")
Output:
[-1/3*(a^3*b^7 - 17*a^4*b^5*c + 118*a^5*b^3*c^2 - 264*a^6*b*c^3 + 3*(b^8*c ^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 236*a^3*b^2*c^5 + 176*a^4*c^6)*x^5 + 3*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 206*a^3*b^3*c^4 + 56*a^4*b*c^5) *x^4 + (b^10 - 11*a*b^8*c + 16*a^2*b^6*c^2 + 334*a^3*b^4*c^3 - 1304*a^4*b^ 2*c^4 + 640*a^5*c^5)*x^3 + 3*(a*b^9 - 16*a^2*b^7*c + 101*a^3*b^5*c^2 - 196 *a^4*b^3*c^3 - 64*a^5*b*c^4)*x^2 + 60*(a^3*c^6*x^6 + 3*a^3*b*c^5*x^5 + 3*a ^5*b*c^3*x + a^6*c^3 + 3*(a^3*b^2*c^4 + a^4*c^5)*x^4 + (a^3*b^3*c^3 + 6*a^ 4*b*c^4)*x^3 + 3*(a^4*b^2*c^3 + a^5*c^4)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2 *x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 3*(a^2*b^8 - 17*a^3*b^6*c + 118*a^4*b^4*c^2 - 284*a^5*b^2*c^3 + 8 0*a^6*c^4)*x)/(a^3*b^8*c^3 - 16*a^4*b^6*c^4 + 96*a^5*b^4*c^5 - 256*a^6*b^2 *c^6 + 256*a^7*c^7 + (b^8*c^6 - 16*a*b^6*c^7 + 96*a^2*b^4*c^8 - 256*a^3*b^ 2*c^9 + 256*a^4*c^10)*x^6 + 3*(b^9*c^5 - 16*a*b^7*c^6 + 96*a^2*b^5*c^7 - 2 56*a^3*b^3*c^8 + 256*a^4*b*c^9)*x^5 + 3*(b^10*c^4 - 15*a*b^8*c^5 + 80*a^2* b^6*c^6 - 160*a^3*b^4*c^7 + 256*a^5*c^9)*x^4 + (b^11*c^3 - 10*a*b^9*c^4 + 320*a^3*b^5*c^6 - 1280*a^4*b^3*c^7 + 1536*a^5*b*c^8)*x^3 + 3*(a*b^10*c^3 - 15*a^2*b^8*c^4 + 80*a^3*b^6*c^5 - 160*a^4*b^4*c^6 + 256*a^6*c^8)*x^2 + 3* (a^2*b^9*c^3 - 16*a^3*b^7*c^4 + 96*a^4*b^5*c^5 - 256*a^5*b^3*c^6 + 256*a^6 *b*c^7)*x), -1/3*(a^3*b^7 - 17*a^4*b^5*c + 118*a^5*b^3*c^2 - 264*a^6*b*c^3 + 3*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 236*a^3*b^2*c^5 + 176*a...
Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (146) = 292\).
Time = 1.72 (sec) , antiderivative size = 938, normalized size of antiderivative = 6.05 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(x**6/(c*x**2+b*x+a)**4,x)
Output:
-20*a**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-5120*a**7*c**4*sqrt(-1/(4*a* c - b**2)**7) + 5120*a**6*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7) - 1920*a**5 *b**4*c**2*sqrt(-1/(4*a*c - b**2)**7) + 320*a**4*b**6*c*sqrt(-1/(4*a*c - b **2)**7) - 20*a**3*b**8*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b)/(40*a**3*c )) + 20*a**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (5120*a**7*c**4*sqrt(-1/(4 *a*c - b**2)**7) - 5120*a**6*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7) + 1920*a **5*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7) - 320*a**4*b**6*c*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b**8*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b)/(40*a** 3*c)) + (66*a**5*b*c**2 - 13*a**4*b**3*c + a**3*b**5 + x**5*(-132*a**3*c** 5 + 144*a**2*b**2*c**4 - 36*a*b**4*c**3 + 3*b**6*c**2) + x**4*(-42*a**3*b* c**4 + 144*a**2*b**3*c**3 - 36*a*b**5*c**2 + 3*b**7*c) + x**3*(-160*a**4*c **4 + 286*a**3*b**2*c**3 - 12*a**2*b**4*c**2 - 7*a*b**6*c + b**8) + x**2*( 48*a**4*b*c**3 + 159*a**3*b**3*c**2 - 36*a**2*b**5*c + 3*a*b**7) + x*(-60* a**5*c**3 + 198*a**4*b**2*c**2 - 39*a**3*b**4*c + 3*a**2*b**6))/(192*a**6* c**6 - 144*a**5*b**2*c**5 + 36*a**4*b**4*c**4 - 3*a**3*b**6*c**3 + x**6*(1 92*a**3*c**9 - 144*a**2*b**2*c**8 + 36*a*b**4*c**7 - 3*b**6*c**6) + x**5*( 576*a**3*b*c**8 - 432*a**2*b**3*c**7 + 108*a*b**5*c**6 - 9*b**7*c**5) + x* *4*(576*a**4*c**8 + 144*a**3*b**2*c**7 - 324*a**2*b**4*c**6 + 99*a*b**6*c* *5 - 9*b**8*c**4) + x**3*(1152*a**4*b*c**7 - 672*a**3*b**3*c**6 + 72*a**2* b**5*c**5 + 18*a*b**7*c**4 - 3*b**9*c**3) + x**2*(576*a**5*c**7 + 144*a...
Exception generated. \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^6/(c*x^2+b*x+a)^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (147) = 294\).
Time = 0.22 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.49 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {40 \, a^{3} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {3 \, b^{6} c^{2} x^{5} - 36 \, a b^{4} c^{3} x^{5} + 144 \, a^{2} b^{2} c^{4} x^{5} - 132 \, a^{3} c^{5} x^{5} + 3 \, b^{7} c x^{4} - 36 \, a b^{5} c^{2} x^{4} + 144 \, a^{2} b^{3} c^{3} x^{4} - 42 \, a^{3} b c^{4} x^{4} + b^{8} x^{3} - 7 \, a b^{6} c x^{3} - 12 \, a^{2} b^{4} c^{2} x^{3} + 286 \, a^{3} b^{2} c^{3} x^{3} - 160 \, a^{4} c^{4} x^{3} + 3 \, a b^{7} x^{2} - 36 \, a^{2} b^{5} c x^{2} + 159 \, a^{3} b^{3} c^{2} x^{2} + 48 \, a^{4} b c^{3} x^{2} + 3 \, a^{2} b^{6} x - 39 \, a^{3} b^{4} c x + 198 \, a^{4} b^{2} c^{2} x - 60 \, a^{5} c^{3} x + a^{3} b^{5} - 13 \, a^{4} b^{3} c + 66 \, a^{5} b c^{2}}{3 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \] Input:
integrate(x^6/(c*x^2+b*x+a)^4,x, algorithm="giac")
Output:
-40*a^3*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2 *b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(3*b^6*c^2*x^5 - 36*a*b^4 *c^3*x^5 + 144*a^2*b^2*c^4*x^5 - 132*a^3*c^5*x^5 + 3*b^7*c*x^4 - 36*a*b^5* c^2*x^4 + 144*a^2*b^3*c^3*x^4 - 42*a^3*b*c^4*x^4 + b^8*x^3 - 7*a*b^6*c*x^3 - 12*a^2*b^4*c^2*x^3 + 286*a^3*b^2*c^3*x^3 - 160*a^4*c^4*x^3 + 3*a*b^7*x^ 2 - 36*a^2*b^5*c*x^2 + 159*a^3*b^3*c^2*x^2 + 48*a^4*b*c^3*x^2 + 3*a^2*b^6* x - 39*a^3*b^4*c*x + 198*a^4*b^2*c^2*x - 60*a^5*c^3*x + a^3*b^5 - 13*a^4*b ^3*c + 66*a^5*b*c^2)/((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^ 6)*(c*x^2 + b*x + a)^3)
Time = 9.20 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.23 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {\frac {a^3\,\left (66\,a^2\,b\,c^2-13\,a\,b^3\,c+b^5\right )}{3\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^5\,\left (-44\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {x^3\,\left (160\,a^4\,c^4-286\,a^3\,b^2\,c^3+12\,a^2\,b^4\,c^2+7\,a\,b^6\,c-b^8\right )}{3\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^4\,\left (-14\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a\,x^2\,\left (16\,a^3\,b\,c^3+53\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a^2\,x\,\left (-20\,a^3\,c^3+66\,a^2\,b^2\,c^2-13\,a\,b^4\,c+b^6\right )}{c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {40\,a^3\,\mathrm {atan}\left (\frac {\left (\frac {20\,a^3\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {40\,a^3\,c\,x}{{\left (4\,a\,c-b^2\right )}^{7/2}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{20\,a^3}\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \] Input:
int(x^6/(a + b*x + c*x^2)^4,x)
Output:
- ((a^3*(b^5 + 66*a^2*b*c^2 - 13*a*b^3*c))/(3*c^3*(b^6 - 64*a^3*c^3 + 48*a ^2*b^2*c^2 - 12*a*b^4*c)) + (x^5*(b^6 - 44*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a *b^4*c))/(c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^3*(160* a^4*c^4 - b^8 + 12*a^2*b^4*c^2 - 286*a^3*b^2*c^3 + 7*a*b^6*c))/(3*c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^4*(b^7 - 14*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/(c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 1 2*a*b^4*c)) + (a*x^2*(b^7 + 16*a^3*b*c^3 + 53*a^2*b^3*c^2 - 12*a*b^5*c))/( c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (a^2*x*(b^6 - 20*a ^3*c^3 + 66*a^2*b^2*c^2 - 13*a*b^4*c))/(c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2 *c^2 - 12*a*b^4*c)))/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3*b^2*c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) - (40*a^3*a tan((((20*a^3*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/((4*a*c - b^2)^(7/2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (40*a^3*c *x)/(4*a*c - b^2)^(7/2))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) /(20*a^3)))/(4*a*c - b^2)^(7/2)
Time = 0.23 (sec) , antiderivative size = 1181, normalized size of antiderivative = 7.62 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int(x^6/(c*x^2+b*x+a)^4,x)
Output:
(120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**6*b*c**2 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**2*c** 2*x + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c **3*x**2 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 4*b**3*c**2*x**2 + 720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* *2))*a**4*b**2*c**3*x**3 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*a**4*b*c**4*x**4 + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq rt(4*a*c - b**2))*a**3*b**4*c**2*x**3 + 360*sqrt(4*a*c - b**2)*atan((b + 2 *c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c**3*x**4 + 360*sqrt(4*a*c - b**2)*ata n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**4*x**5 + 120*sqrt(4*a*c - b **2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**5*x**6 + 176*a**7*c**3 + 28*a**6*b**2*c**2 + 288*a**6*b*c**3*x + 528*a**6*c**4*x**2 - 22*a**5*b* *4*c + 144*a**5*b**3*c**2*x + 12*a**5*b**2*c**3*x**2 + 416*a**5*b*c**4*x** 3 + 528*a**5*c**5*x**4 + a**4*b**6 - 66*a**4*b**5*c*x + 168*a**4*b**4*c**2 *x**2 + 64*a**4*b**3*c**3*x**3 - 348*a**4*b**2*c**4*x**4 + 176*a**4*c**6*x **6 + 3*a**3*b**7*x - 63*a**3*b**6*c*x**2 + 6*a**3*b**5*c**2*x**3 + 198*a* *3*b**4*c**3*x**4 - 236*a**3*b**2*c**5*x**6 + 3*a**2*b**8*x**2 - 16*a**2*b **7*c*x**3 - 48*a**2*b**6*c**2*x**4 + 96*a**2*b**4*c**4*x**6 + a*b**9*x**3 + 3*a*b**8*c*x**4 - 16*a*b**6*c**3*x**6 + b**8*c**2*x**6)/(3*b*c**2*(256* a**7*c**4 - 256*a**6*b**2*c**3 + 768*a**6*b*c**4*x + 768*a**6*c**5*x**2...